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4.9H — The Arithmetic of Matrices
Human review completed .
Choose the appropriate matrix operation, calculate selected entries, and recognize an identity matrix. The packet labels this section 4.9H.
1. In class: A Practice Understanding Task
Choose the operation
For a refresher on addition and scaling, see Section 4.7H; for row-by-column multiplication, see Section 4.8H.
| Operation | What to do |
|---|---|
| Add or subtract matrices | Combine corresponding entries. The matrices must have the same dimensions. Subtraction means subtracting each second-matrix entry from its matching first-matrix entry. |
| Multiply by a number | Multiply every entry by that number. |
| Multiply two matrices | Use one row from the first and one column from the second. Multiply matching entries, then add the products. The inner dimensions must match. |
Volunteer hours and credits — Main 1–3
The packet gives September volunteer hours for drama and chess. Both matrices have rows for tables, floors, dishes and columns for weekdays, weekends:
1 — Combined hours: Add corresponding entries:
2 — Nine months of drama hours: Multiply every September entry by 9:
3 — Chess club’s September credit by chore: Multiply the chess hours by the rate column, which gives $4 per weekday hour and $6 per weekend hour:
For example, tables earn \(5(4)+2(6)=32\) dollars. The product has one column of dollar credits, with the rows still tables, floors, and dishes.
Worked example — Find just one product entry
Original problem — Main 5: Find the boxed entries of the packet’s product. Consider the box in row 1, column 2.
Worked solution: The first matrix’s row 1 is \((-2,3)\); the second matrix’s column 2 contains −3 and 5. Use only those values:
There is no need to calculate the other entries to find this one.
Selected-entry answers — Main 4–6
Rows and columns are counted from the top left, starting at 1. These are only the boxed entries requested by the packet:
| Main problem | Box location | Answer |
|---|---|---|
| 4 | Row 1, column 3 | 8 |
| 4 | Row 2, column 2 | −2 |
| 5 | Row 1, column 2 | 21 |
| 5 | Row 2, column 1 | 9 |
| 5 | Row 3, column 3 | −2 |
| 6 | Row 1, column 2 | 24 |
| 6 | Row 2, column 1 | 17 |
In Main 6, apply the scalar factors before subtracting matching entries. For the upper-right box, \(3(4)-4(-3)=12-(-12)=24\).
The identity matrix — Main 7
An identity matrix is a square matrix with 1s on its diagonal from top left to bottom right, and 0s elsewhere. For three rows and columns:
Multiplying a matrix by a matching identity matrix leaves it unchanged. Each identity row selects the corresponding row of the other matrix: for example, the first row gives \(1\times\text{row 1}+0\times\text{row 2}+0\times\text{row 3}\).
To solve Main 7, first find the blank in the left matrix. Call it q. The product’s third-row, second-column entry is 3, so:
The left matrix is therefore I. The right matrix and the product must be identical. Fill the remaining blanks by matching their positions:
| Matrix | Blank location | Answer |
|---|---|---|
| Left matrix | Row 3, column 2 | 0 |
| Right matrix | Row 1, column 3 | 5 |
| Right matrix | Row 3, column 1 | 4 |
| Product | Row 1, column 2 | 3 |
| Product | Row 2, column 3 | 4 |
| Product | Row 3, column 3 | 1 |
2. Ready — Graph linear relationships
For a refresher, see Section 3.2’s line-graphing examples. Choose two points far enough apart for an accurate line. Try your graphs before opening the key.
Check your work — Ready 1–5 answer key
Plots are not shown. The rules and sample point pairs below let you check the lines you draw. Other points on the same lines are valid.
| Problem | Line | Two points |
|---|---|---|
| 1 | \(y=3x-5\) | \((-1,-8)\) and \((5,10)\) |
| 2 | \(y=2x+7\) | \((-5,-3)\) and \((1,9)\) |
| 3 | \(y=-2x\) | \((-5,10)\) and \((5,-10)\) |
| 4 | \(y=\frac15x\) | \((-10,-2)\) and \((10,2)\) |
| 5 | \(y=6x-7\) | \((0,-7)\) and \((2,5)\) |
3. Set — Matrix arithmetic
Use the operation reminder above. Try each calculation before opening the key. For products, check the dimensions and use row-by-column multiplication.
Check your work — Set 6–13 answer key
6
7
8
9
10
11
12
13
4. Go — Evaluate expressions
This reviews substitution and the order of operations. Use \(x=7\), \(y=-3\), and \(z=5\), then check your results in the key.
Check your work — Go 14–19 answer key
\(-13\)
\(28\)
\(-2\)
\(5929\)
\(\frac23\)
\(-54\)
Additional learning resources
| Resource | What it helps with |
|---|---|
| Khan Academy — Intro to matrix multiplication | A refresher on the row-by-column method used in Main and Set. |
| Khan Academy — Matrix addition and subtraction | Combining matching entries and working with negative entries. |
Teacher’s review of the source packet
No substantial mathematical issues were identified in this section.
Coverage
| Packet component | Where supported |
|---|---|
| Main 1–3 · pp. 41–42 | Operation reminder; visible matrix answers with units and a credit calculation. |
| Main 4–6 · pp. 42–43 | One selected-entry example and answers for the requested boxes only. |
| Main 7 · p. 43 | Identity-matrix definition, determination of the left blank, and all six missing entries. |
| Ready 1–5 · p. 44 | Linked graphing review; rules and separated point pairs. Plots are not reproduced. |
| Set 6–13 · p. 45 | Complete matrix-result key. |
| Go 14–19 · p. 45 | Brief review and numerical answers only. |
Data and prompts adapted from Mathematics Vision Project, Secondary Mathematics I, Module 4 student packet, Section 4.9H, pp. 41–45, under CC BY 4.0. Explanations and solutions added for this supplement.
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